Question

If the farmer has 162 feet of fencing, what are the dimensions of the region which enclose the maximal area

Answer 1

Answer:

The region is a square with a length side of 40.5 feet

Step-by-step explanation:

I will assume that the region is a rectangular area

we know that

The perimeter of the region is equal to

$P=2(x+y)$

where

x is the length

y is the width

we have

$P=162\ ft$

so

$162=2(x+y)$

simplify

$81=x+y$

$y=81-x$ ----> equation A

The area of the rectangular region is

$A=xy$ ----> substitute equation A in equation B

$A=x(81-x)$

$A=-x^2+81x$

This is the equation of a vertical parabola open downward

The vertex represent a maximum

The y-coordinate of the vertex represent the maximum area

The x-coordinate of the vertex represent the length for the maximum area

Convert the quadratic equation in vertex form

Factor -1

$A=-(x^2-81x)$

Complete the square

$A=-(x^2-81x+40.5^2)+40.5^2$

$A=-(x^2-81x+40.5^2)+1,640.25$

Rewrite as perfect squares

$A=-(x-40.5)^2+1,640.25$

The vertex is the point (40.5,1,640.25)

so

The maximum area is $1.640.25\ ft^2$

The length for the maximum area is $x=40.5\ ft$

Find the value of y (equation A)

$y=81-40.5=40.5\ ft$

therefore

The region is a square with a length side of 40.5 feet